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What is CZM and why is it important
In the study of solids and design of nano/micro/macro structures,
thermomechanical behavior is modeled through constitutive equations.
(Barenblatt, G.I, (1959), PMM (23) p. 434)
Dugdale (1960)
independently developed the concept of cohesive stress
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➢ For Ductile metals (steel)
➢ Cohesive stress in the CZM is equated to yield stress Y
Development of CZ Models-Historical Review
Figure (a) Variation of Cohesive traction (b) I - inner region, II - edge region
Barenblatt (1959) was
first to propose the concept of Cohesive zone model to brittle fracture
C
y B
NO MATERIAL SEPARATION
A
l1
d max
FORWARD
D
LOCATION OF COHESIVE CRACK TIP
d D
l2
WAKE
COMPLETE MATERIAL SEPARATION
E d, X
d sep
MATERIAL CRACK TIP
COHESIVE CRACK TIP
practical problems with use of FEM and advent of fast computing. Model has been recast as a phenomenological one for a number of systems and
the atomic scale. ➢ It can also be perceived at the meso- scale as the
effect of energy dissipation mechanisms, energy dissipated both in the forward and the wake regions of the crack tip. ➢ Uses fracture energy(obtained from fracture tests) as a parameter and is devoid of any ad-hoc criteria for fracture initiation and propagation. ➢ Eliminates singularity of stress and limits it to the cohesive strength of the the material. ➢ It is an ideal framework to model strength, stiffness and failure in an integrated manner. ➢ Applications: geomaterials, biomaterials, concrete, metallics, composites….
➢ Analyzed for plastic zone size for plates under tension
➢ Length of yielding zone ‘s’, theoretical crack length ‘a’, and applied loading ‘T’ are related in
MATHEMATICAL CRACK TIP
INACTIVE PLASTIC ZONE (Plastic wake)
d sep
dD
d max
A
E
D
C
WAKE
FORWARD
y ACTIVE PLASTIC ZONE
x
ELASTIC SINGULARITY ZONE
Concept of wake and forward region in the cohesive process zone
➢ Molecular force of cohesion acting near the edge of the crack at its surface (region II ). ➢ The intensity of molecular force of cohesion ‘f ’ is found to vary as shown in Fig.a. ➢ The interatomic force is initially zero when the atomic planes are separated by normal
Typically is a continuous function of , , f(, , ) and their history. Design is limited by a maximum value of a given parameter () at any local point.
After fracture the surface 1 comprise of unseparated surface and completely separated surface (e.g. * ); all modeled within the con-
cept of CZM. Such an approach is not possible in conventional mechanics of continuous media.
Wake of crack tip
Forward of crack tip
Fibril (MMC bridging Grain bridging
Microvoid coalescence
Plastic zone
Metallic
Cleavage fracture
Oxide bridging
Fibril(polymers) bridging
unique ➢ Additional criteria are required for crack
initiation and propagation
Basic breakdown of the principles of mechanics of continuous media
Damage mechanics-
Plastic wake
Thickness of ceramic interface Crack Meandering
Ceramic
Intrinsic dissipation
Crack Deflection
Precipitates
Extrinsic dissipation
Micro cracking initiation
intermolecular distance and increases to high maximum fm ETo / b E /10 after that it rapidly reduces to zero with increase in separation distance.
E is Young’s modulus and Tois surface tension
➢ can effectively reduce the strength and stiffness of the material in an average sense, but cannot create new surface
D 1 E , Effective stress =
E
1 D
the form
s a
2
sin
2
(
4
T Y
)
(Dugdale, D.S. (1960), J. Mech.Phys.Solids,8,p.100)
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Phenomenological Models
The theory of CZM is based on sound principles. However implementation of model for practical problems grew exponentially for
Face centered atoms
Cyclic load induced crack closure
Phase transformation
Inter/trans granular fracture
BCC
Corner atoms
Body centered atoms
AMML Active dissipation mechanisims participating at the cohesive process zone
Dissipative Micromechanisims Acting in the wake and forward region of the process zone at the Interfaces of Monolithic and Heterogeneous Material
ˆ
max
Fracture/Damage theories to model failure